微分熵

此词条暂由Henry翻译。 由CecileLi初步审校。

Differential entropy (also referred to as continuous entropy) is a concept in information theory that began as an attempt by Shannon to extend the idea of (Shannon) entropy, a measure of average surprisal of a random variable, to continuous probability distributions. Unfortunately, Shannon did not derive this formula, and rather just assumed it was the correct continuous analogue of discrete entropy, but it is not. The actual continuous version of discrete entropy is the limiting density of discrete points (LDDP). Differential entropy (described here) is commonly encountered in the literature, but it is a limiting case of the LDDP, and one that loses its fundamental association with discrete entropy.

Differential entropy (also referred to as continuous entropy) is a concept in information theory that began as an attempt by Shannon to extend the idea of (Shannon) entropy, a measure of average surprisal of a random variable, to continuous probability distributions. Unfortunately, Shannon did not derive this formula, and rather just assumed it was the correct continuous analogue of discrete entropy, but it is not.

微分熵Differential entropy(也被称为连续熵)是信息论中的一个概念，其来源于香农尝试将他的香农熵的概念扩展到连续的概率分布。香农熵是衡量一个随机变量的平均惊异程度的指标。可惜的是，香农只是假设它是离散熵的正确连续模拟而并没有推导出公式，但事实上它并不是离散熵的正确连续模拟。

[math]\displaystyle{ h(X_1, \ldots, X_n) = \sum_{i=1}^{n} h(X_i|X_1, \ldots, X_{i-1}) \leq \sum_{i=1}^{n} h(X_i) }[/math].

Definition

定义 Let [math]\displaystyle{ X }[/math] be a random variable with a probability density function [math]\displaystyle{ f }[/math] whose support is a set [math]\displaystyle{ \mathcal X }[/math]. The differential entropy [math]\displaystyle{ h(X) }[/math] or [math]\displaystyle{ h(f) }[/math] is defined as [math]\displaystyle{ h(X+c) = h(X) }[/math]

 --CecileLi(讨论)  【审校】此处缺无格式的英文及翻译 补充：设随机变量X，其概率密度函数F的的定义域是X的集合

[math]\displaystyle{ h(X) = -\int_\mathcal{X} f(x)\log f(x)\,dx }[/math]

For probability distributions which don't have an explicit density function expression, but have an explicit quantile function expression, [math]\displaystyle{ Q(p) }[/math], then [math]\displaystyle{ h(Q) }[/math] can be defined in terms of the derivative of [math]\displaystyle{ Q(p) }[/math] i.e. the quantile density function [math]\displaystyle{ Q'(p) }[/math] as

--CecileLi(讨论)  【审校】此处缺无格式的英文及翻译 补充：For probability distributions which don't have an explicit density function expression, but have an explicit quantile function expression, , then  can be defined in terms of the derivative of  i.e. the quantile density function as

对于没有显式密度函数表达式，但有显式分位数函数表达式的概率分布，我们则可以用分位数密度函数的导数来定义，即

[math]\displaystyle{ h(Q) = \int_0^1 \log Q'(p)\,dp }[/math].

A modification of differential entropy that addresses these drawbacks is the relative information entropy, also known as the Kullback–Leibler divergence, which includes an invariant measure factor (see limiting density of discrete points).

针对这些缺点的一个改进方法是相对熵，也被称为 Kullback-Leibler 分歧，其中包括一个不变测度因子(见离散点的极限密度)。

As with its discrete analog, the units of differential entropy depend on the base of the logarithm, which is usually 2 (i.e., the units are bits). See logarithmic units for logarithms taken in different bases. Related concepts such as joint, conditional differential entropy, and relative entropy are defined in a similar fashion. Unlike the discrete analog, the differential entropy has an offset that depends on the units used to measure [math]\displaystyle{ X }[/math]. For example, the differential entropy of a quantity measured in millimeters will be 模板:Not a typo more than the same quantity measured in meters; a dimensionless quantity will have differential entropy of 模板:Not a typo more than the same quantity divided by 1000.

One must take care in trying to apply properties of discrete entropy to differential entropy, since probability density functions can be greater than 1. For example, the uniform distribution [math]\displaystyle{ \mathcal{U}(0,1/2) }[/math] has negative differential entropy

[math]\displaystyle{ \int_0^\frac{1}{2} -2\log(2)\,dx=-\log(2)\, }[/math].

Thus, differential entropy does not share all properties of discrete entropy.

Note that the continuous mutual information [math]\displaystyle{ I(X;Y) }[/math] has the distinction of retaining its fundamental significance as a measure of discrete information since it is actually the limit of the discrete mutual information of partitions of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] as these partitions become finer and finer. Thus it is invariant under non-linear homeomorphisms (continuous and uniquely invertible maps), including linear transformations of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math], and still represents the amount of discrete information that can be transmitted over a channel that admits a continuous space of values.

For the direct analogue of discrete entropy extended to the continuous space, see limiting density of discrete points.

Properties of differential entropy

微分熵的性质

• For probability densities [math]\displaystyle{ f }[/math] and [math]\displaystyle{ g }[/math], the Kullback–Leibler divergence [math]\displaystyle{ D_{KL}(f || g) }[/math] is greater than or equal to 0 with equality only if [math]\displaystyle{ f=g }[/math] almost everywhere. Similarly, for two random variables [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math], [math]\displaystyle{ I(X;Y) \ge 0 }[/math] and [math]\displaystyle{ h(X|Y) \le h(X) }[/math] with equality if and only if [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are independent.

• The chain rule for differential entropy holds as in the discrete case

[math]\displaystyle{ h(X_1, \ldots, X_n) = \sum_{i=1}^{n} h(X_i|X_1, \ldots, X_{i-1}) \leq \sum_{i=1}^{n} h(X_i) }[/math].

• Differential entropy is translation invariant, i.e. for a constant [math]\displaystyle{ c }[/math].

[math]\displaystyle{ h(X+c) = h(X) }[/math]

• Differential entropy is in general not invariant under arbitrary invertible maps.

In particular, for a constant [math]\displaystyle{ a }[/math]

[math]\displaystyle{ 0=\delta L=\int_{-\infty}^\infty \delta g(x)\left (\ln(g(x))+1+\lambda_0+\lambda(x-\mu)^2\right )\,dx }[/math]

[math]\displaystyle{ h(aX) = h(X)+ \log |a| }[/math]

For a vector valued random variable [math]\displaystyle{ \mathbf{X} }[/math] and an invertible (square) matrix [math]\displaystyle{ \mathbf{A} }[/math]

[math]\displaystyle{ h(\mathbf{A}\mathbf{X})=h(\mathbf{X})+\log \left( |\det \mathbf{A}| \right) }[/math]

• In general, for a transformation from a random vector to another random vector with same dimension [math]\displaystyle{ \mathbf{Y}=m \left(\mathbf{X}\right) }[/math], the corresponding entropies are related via

[math]\displaystyle{ h(\mathbf{Y}) \leq h(\mathbf{X}) + \int f(x) \log \left\vert \frac{\partial m}{\partial x} \right\vert dx }[/math]

where [math]\displaystyle{ \left\vert \frac{\partial m}{\partial x} \right\vert }[/math] is the Jacobian of the transformation [math]\displaystyle{ m }[/math]. The above inequality becomes an equality if the transform is a bijection. Furthermore, when [math]\displaystyle{ m }[/math] is a rigid rotation, translation, or combination thereof, the Jacobian determinant is always 1, and [math]\displaystyle{ h(Y)=h(X) }[/math].

• If a random vector [math]\displaystyle{ X \in \mathbb{R}^n }[/math] has mean zero and covariance matrix [math]\displaystyle{ K }[/math], [math]\displaystyle{ h(\mathbf{X}) \leq \frac{1}{2} \log(\det{2 \pi e K}) = \frac{1}{2} \log[(2\pi e)^n \det{K}] }[/math] with equality if and only if [math]\displaystyle{ X }[/math] is jointly gaussian (see below).

• It is not invariant under change of variables, and is therefore most useful with dimensionless variables.

它在变量变化下不是不变的，因此对无量纲变量最有用

• It can be negative.

它可以为负

A modification of differential entropy that addresses these drawbacks is the relative information entropy, also known as the Kullback–Leibler divergence, which includes an invariant measure factor (see limiting density of discrete points).

Maximization in the normal distribution

正态分布中的最大化

Theorem

理论 Its differential entropy is then 它的微分熵就会 With a normal distribution, differential entropy is maximized for a given variance. A Gaussian random variable has the largest entropy amongst all random variables of equal variance, or, alternatively, the maximum entropy distribution under constraints of mean and variance is the Gaussian. 对于正态分布，对于给定的方差，微分熵是最大的。在所有等方差随机变量中，高斯随机变量的熵最大，或者在均值和方差约束下的最大熵分布是高斯分布

Proof

证明

Let [math]\displaystyle{ g(x) }[/math] be a Gaussian PDF with mean μ and variance [math]\displaystyle{ \sigma^2 }[/math] and [math]\displaystyle{ f(x) }[/math] an arbitrary PDF with the same variance. Since differential entropy is translation invariant we can assume that [math]\displaystyle{ f(x) }[/math] has the same mean of [math]\displaystyle{ \mu }[/math] as [math]\displaystyle{ g(x) }[/math].

Consider the Kullback–Leibler divergence between the two distributions

[math]\displaystyle{ 0 \leq D_{KL}(f || g) = \int_{-\infty}^\infty f(x) \log \left( \frac{f(x)}{g(x)} \right) dx = -h(f) - \int_{-\infty}^\infty f(x)\log(g(x)) dx. }[/math]

Now note that

[math]\displaystyle{ \begin{align} \int_{-\infty}^\infty f(x)\log(g(x)) dx &= \int_{-\infty}^\infty f(x)\log\left( \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\right) dx \\ &= \int_{-\infty}^\infty f(x) \log\frac{1}{\sqrt{2\pi\sigma^2}} dx + \log(e)\int_{-\infty}^\infty f(x)\left( -\frac{(x-\mu)^2}{2\sigma^2}\right) dx \\ &= -\tfrac{1}{2}\log(2\pi\sigma^2) - \log(e)\frac{\sigma^2}{2\sigma^2} \\ &= -\tfrac{1}{2}\left(\log(2\pi\sigma^2) + \log(e)\right) \\ &= -\tfrac{1}{2}\log(2\pi e \sigma^2) \\ &= -h(g) \end{align} }[/math]

because the result does not depend on [math]\displaystyle{ f(x) }[/math] other than through the variance. Combining the two results yields

[math]\displaystyle{ h(g) - h(f) \geq 0 \! }[/math]

with equality when [math]\displaystyle{ f(x)=g(x) }[/math] following from the properties of Kullback–Leibler divergence.

Alternative proof

替代证明

This result may also be demonstrated using the variational calculus. A Lagrangian function with two Lagrangian multipliers may be defined as:

[math]\displaystyle{ L=\int_{-\infty}^\infty g(x)\ln(g(x))\,dx-\lambda_0\left(1-\int_{-\infty}^\infty g(x)\,dx\right)-\lambda\left(\sigma^2-\int_{-\infty}^\infty g(x)(x-\mu)^2\,dx\right) }[/math]

where g(x) is some function with mean μ. When the entropy of g(x) is at a maximum and the constraint equations, which consist of the normalization condition [math]\displaystyle{ \left(1=\int_{-\infty}^\infty g(x)\,dx\right) }[/math] and the requirement of fixed variance [math]\displaystyle{ \left(\sigma^2=\int_{-\infty}^\infty g(x)(x-\mu)^2\,dx\right) }[/math], are both satisfied, then a small variation δg(x) about g(x) will produce a variation δL about L which is equal to zero:

[math]\displaystyle{ 0=\delta L=\int_{-\infty}^\infty \delta g(x)\left (\ln(g(x))+1+\lambda_0+\lambda(x-\mu)^2\right )\,dx }[/math]

Since this must hold for any small δg(x), the term in brackets must be zero, and solving for g(x) yields:

[math]\displaystyle{ g(x)=e^{-\lambda_0-1-\lambda(x-\mu)^2} }[/math]

Using the constraint equations to solve for λ₀ and λ yields the normal distribution:

[math]\displaystyle{ g(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} }[/math]

Example: Exponential distribution

例子：指数分布 Let [math]\displaystyle{ X }[/math] be an exponentially distributed random variable with parameter [math]\displaystyle{ \lambda }[/math], that is, with probability density function

[math]\displaystyle{ f(x) = \lambda e^{-\lambda x} \mbox{ for } x \geq 0. }[/math]

Its differential entropy is then

[math]\displaystyle{ h_e(X)\, }[/math]	[math]\displaystyle{ =-\int_0^\infty \lambda e^{-\lambda x} \log (\lambda e^{-\lambda x})\,dx }[/math]
	[math]\displaystyle{ = -\left(\int_0^\infty (\log \lambda)\lambda e^{-\lambda x}\,dx + \int_0^\infty (-\lambda x) \lambda e^{-\lambda x}\,dx\right) }[/math]
	[math]\displaystyle{ = -\log \lambda \int_0^\infty f(x)\,dx + \lambda E[X] }[/math]
	[math]\displaystyle{ = -\log\lambda + 1\,. }[/math]

Here, [math]\displaystyle{ h_e(X) }[/math] was used rather than [math]\displaystyle{ h(X) }[/math] to make it explicit that the logarithm was taken to base e, to simplify the calculation.

Relation to estimator error

The differential entropy yields a lower bound on the expected squared error of an estimator. For any random variable [math]\displaystyle{ X }[/math] and estimator [math]\displaystyle{ \widehat{X} }[/math] the following holds:

[math]\displaystyle{ \operatorname{E}[(X - \widehat{X})^2] \ge \frac{1}{2\pi e}e^{2h(X)} }[/math]

with equality if and only if [math]\displaystyle{ X }[/math] is a Gaussian random variable and [math]\displaystyle{ \widehat{X} }[/math] is the mean of [math]\displaystyle{ X }[/math].

Differential entropies for various distributions

In the table below [math]\displaystyle{ \Gamma(x) = \int_0^{\infty} e^{-t} t^{x-1} dt }[/math] is the gamma function, [math]\displaystyle{ \psi(x) = \frac{d}{dx} \ln\Gamma(x)=\frac{\Gamma'(x)}{\Gamma(x)} }[/math] is the digamma function, [math]\displaystyle{ B(p,q) = \frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)} }[/math] is the beta function, and γ_E is Euler's constant.

Table of differential entropies

Distribution Name	Probability density function (pdf)	Entropy in nats	Support
Uniform	[math]\displaystyle{ f(x) = \frac{1}{b-a} }[/math]	[math]\displaystyle{ \ln(b - a) \, }[/math]	[math]\displaystyle{ [a,b]\, }[/math]
Normal	[math]\displaystyle{ f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) }[/math]	[math]\displaystyle{ \ln\left(\sigma\sqrt{2\,\pi\,e}\right) }[/math]	[math]\displaystyle{ (-\infty,\infty)\, }[/math]
Exponential	[math]\displaystyle{ f(x) = \lambda \exp\left(-\lambda x\right) }[/math]	[math]\displaystyle{ 1 - \ln \lambda \, }[/math]	[math]\displaystyle{ [0,\infty)\, }[/math]
Rayleigh	[math]\displaystyle{ f(x) = \frac{x}{\sigma^2} \exp\left(-\frac{x^2}{2\sigma^2}\right) }[/math]	[math]\displaystyle{ 1 + \ln \frac{\sigma}{\sqrt{2}} + \frac{\gamma_E}{2} }[/math]	[math]\displaystyle{ [0,\infty)\, }[/math]
Beta	[math]\displaystyle{ f(x) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)} }[/math] for [math]\displaystyle{ 0 \leq x \leq 1 }[/math]	[math]\displaystyle{ \ln B(\alpha,\beta) - (\alpha-1)[\psi(\alpha) - \psi(\alpha +\beta)]\, }[/math] [math]\displaystyle{ - (\beta-1)[\psi(\beta) - \psi(\alpha + \beta)] \, }[/math]	[math]\displaystyle{ [0,1]\, }[/math]
Cauchy	[math]\displaystyle{ f(x) = \frac{\gamma}{\pi} \frac{1}{\gamma^2 + x^2} }[/math]	[math]\displaystyle{ \ln(4\pi\gamma) \, }[/math]	[math]\displaystyle{ (-\infty,\infty)\, }[/math]
Chi	[math]\displaystyle{ f(x) = \frac{2}{2^{k/2} \Gamma(k/2)} x^{k-1} \exp\left(-\frac{x^2}{2}\right) }[/math]	[math]\displaystyle{ \ln{\frac{\Gamma(k/2)}{\sqrt{2}}} - \frac{k-1}{2} \psi\left(\frac{k}{2}\right) + \frac{k}{2} }[/math]	[math]\displaystyle{ [0,\infty)\, }[/math]
Chi-squared	[math]\displaystyle{ f(x) = \frac{1}{2^{k/2} \Gamma(k/2)} x^{\frac{k}{2}\!-\!1} \exp\left(-\frac{x}{2}\right) }[/math]	[math]\displaystyle{ \ln 2\Gamma\left(\frac{k}{2}\right) - \left(1 - \frac{k}{2}\right)\psi\left(\frac{k}{2}\right) + \frac{k}{2} }[/math]	[math]\displaystyle{ [0,\infty)\, }[/math]
Erlang	[math]\displaystyle{ f(x) = \frac{\lambda^k}{(k-1)!} x^{k-1} \exp(-\lambda x) }[/math]	[math]\displaystyle{ (1-k)\psi(k) + \ln \frac{\Gamma(k)}{\lambda} + k }[/math]	[math]\displaystyle{ [0,\infty)\, }[/math]
F	[math]\displaystyle{ f(x) = \frac{n_1^{\frac{n_1}{2}} n_2^{\frac{n_2}{2}}}{B(\frac{n_1}{2},\frac{n_2}{2})} \frac{x^{\frac{n_1}{2} - 1}}{(n_2 + n_1 x)^{\frac{n_1 + n2}{2}}} }[/math]	[math]\displaystyle{ \ln \frac{n_1}{n_2} B\left(\frac{n_1}{2},\frac{n_2}{2}\right) + \left(1 - \frac{n_1}{2}\right) \psi\left(\frac{n_1}{2}\right) - }[/math] [math]\displaystyle{ \left(1 + \frac{n_2}{2}\right)\psi\left(\frac{n_2}{2}\right) + \frac{n_1 + n_2}{2} \psi\left(\frac{n_1\!+\!n_2}{2}\right) }[/math]	[math]\displaystyle{ [0,\infty)\, }[/math]
Gamma	[math]\displaystyle{ f(x) = \frac{x^{k - 1} \exp(-\frac{x}{\theta})}{\theta^k \Gamma(k)} }[/math]	[math]\displaystyle{ \ln(\theta \Gamma(k)) + (1 - k)\psi(k) + k \, }[/math]	[math]\displaystyle{ [0,\infty)\, }[/math]
Laplace	[math]\displaystyle{ f(x) = \frac{1}{2b} \exp\left(-\frac{|x - \mu|}{b}\right) }[/math]	[math]\displaystyle{ 1 + \ln(2b) \, }[/math]	[math]\displaystyle{ (-\infty,\infty)\, }[/math]
Logistic	[math]\displaystyle{ f(x) = \frac{e^{-x}}{(1 + e^{-x})^2} }[/math]	[math]\displaystyle{ 2 \, }[/math]	[math]\displaystyle{ (-\infty,\infty)\, }[/math]
Lognormal	[math]\displaystyle{ f(x) = \frac{1}{\sigma x \sqrt{2\pi}} \exp\left(-\frac{(\ln x - \mu)^2}{2\sigma^2}\right) }[/math]	[math]\displaystyle{ \mu + \frac{1}{2} \ln(2\pi e \sigma^2) }[/math]	[math]\displaystyle{ [0,\infty)\, }[/math]
Maxwell–Boltzmann	[math]\displaystyle{ f(x) = \frac{1}{a^3}\sqrt{\frac{2}{\pi}}\,x^{2}\exp\left(-\frac{x^2}{2a^2}\right) }[/math]	[math]\displaystyle{ \ln(a\sqrt{2\pi})+\gamma_E-\frac{1}{2} }[/math]	[math]\displaystyle{ [0,\infty)\, }[/math]
Generalized normal	[math]\displaystyle{ f(x) = \frac{2 \beta^{\frac{\alpha}{2}}}{\Gamma(\frac{\alpha}{2})} x^{\alpha - 1} \exp(-\beta x^2) }[/math]	[math]\displaystyle{ \ln{\frac{\Gamma(\alpha/2)}{2\beta^{\frac{1}{2}}}} - \frac{\alpha - 1}{2} \psi\left(\frac{\alpha}{2}\right) + \frac{\alpha}{2} }[/math]	[math]\displaystyle{ (-\infty,\infty)\, }[/math]
Pareto	[math]\displaystyle{ f(x) = \frac{\alpha x_m^\alpha}{x^{\alpha+1}} }[/math]	[math]\displaystyle{ \ln \frac{x_m}{\alpha} + 1 + \frac{1}{\alpha} }[/math]	[math]\displaystyle{ [x_m,\infty)\, }[/math]
Student's t	[math]\displaystyle{ f(x) = \frac{(1 + x^2/\nu)^{-\frac{\nu+1}{2}}}{\sqrt{\nu}B(\frac{1}{2},\frac{\nu}{2})} }[/math]	[math]\displaystyle{ \frac{\nu\!+\!1}{2}\left(\psi\left(\frac{\nu\!+\!1}{2}\right)\!-\!\psi\left(\frac{\nu}{2}\right)\right)\!+\!\ln \sqrt{\nu} B\left(\frac{1}{2},\frac{\nu}{2}\right) }[/math]	[math]\displaystyle{ (-\infty,\infty)\, }[/math]
Triangular	[math]\displaystyle{ f(x) = \begin{cases} \frac{2(x-a)}{(b-a)(c-a)} & \mathrm{for\ } a \le x \leq c, \\[4pt] \frac{2(b-x)}{(b-a)(b-c)} & \mathrm{for\ } c \lt x \le b, \\[4pt] \end{cases} }[/math]	[math]\displaystyle{ \frac{1}{2} + \ln \frac{b-a}{2} }[/math]	[math]\displaystyle{ [0,1]\, }[/math]
Weibull	[math]\displaystyle{ f(x) = \frac{k}{\lambda^k} x^{k-1} \exp\left(-\frac{x^k}{\lambda^k}\right) }[/math]	[math]\displaystyle{ \frac{(k-1)\gamma_E}{k} + \ln \frac{\lambda}{k} + 1 }[/math]	[math]\displaystyle{ [0,\infty)\, }[/math]
Multivariate normal	[math]\displaystyle{ f_X(\vec{x}) = }[/math] [math]\displaystyle{ \frac{\exp \left( -\frac{1}{2} ( \vec{x} - \vec{\mu})^\top \Sigma^{-1}\cdot(\vec{x} - \vec{\mu}) \right)} {(2\pi)^{N/2} \left|\Sigma\right|^{1/2}} }[/math]	[math]\displaystyle{ \frac{1}{2}\ln\{(2\pi e)^{N} \det(\Sigma)\} }[/math]	[math]\displaystyle{ \mathbb{R}^N }[/math]

Many of the differential entropies are from.

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